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\usepackage{amsmath, amsthm, amssymb, bm} % 数学公式与符号
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\usepackage{pythonhighlight}
\usepackage{url} 
\usepackage{enumitem}

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%%文档的题目、作者与日期
\author{2024级数学与应用数学1班}
\title{常微分方程期中考试}
\date{2025年11月13日}

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\begin{document}

\maketitle

% \noindent\textbf{考试说明：} 本试卷共10题，满分100分，考试时间120分钟。请将答案写在答题纸上，写在试卷上无效。

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\begin{enumerate}
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\item  验证函数 $y=C_1\cos(2x)+C_2\sin(2x)$ 是微分方程 $y''+4y=0$ 的解。

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\item  求解初值问题：$y''=e^x, \, y(0)=1, y'(0)=2$。

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\item  求解微分方程：$\frac{dy}{dx} = \frac{y}{x}$，并指出在平面上有意义的区域。

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\item  求解微分方程：$\frac{dy}{dx} - y = e^{2x}$。

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\item  求解微分方程：$y' = \frac{y+x}{y-x}$。

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\item  将自治的二阶微分方程 $y''-2y'+y=0$ 化为一阶微分方程组。

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\item  求齐次线性微分方程组 $\frac{d \vec{y}}{dt}=A(t)\vec{y}$ 的通解，其中 $A(t) = \begin{bmatrix} 0 & \frac{1}{t} \\ -\frac{1}{t} & 0 \end{bmatrix}$。

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\item  设 $A=\begin{bmatrix} 1 & 0 \\  1 & 1 \end{bmatrix}$, 求 $\exp(A)$。

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\item  求解一阶线性方程在原点附近的幂级数解：$y' - xy = 0$。

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\item  作出平面动力系统在原点附近的相图：
\begin{eqnarray*}
\left\{ \begin{array}{rcl}
\frac{dx}{dt} &=& x+y, \\
\frac{dy}{dt} &=& -x+y.\\
\end{array}\right.
\end{eqnarray*}

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\end{enumerate}

\end{document}